Chapter 20: The Hull-White Model in Detail¶
This chapter provides a comprehensive treatment of the Hull-White extended Vasicek model for interest rate derivatives, from its mathematical foundations through analytical bond and derivative pricing, tree-based and Monte Carlo implementations, and calibration to market data. Starting from the HJM framework and the connection to Vasicek, we derive the model's affine structure, solve for the short rate and bond prices in closed form, develop the named function apparatus, price bond options, caps, floors, and swaptions analytically, and extend to the two-factor model for richer yield curve dynamics.
Key Concepts¶
Model Definition and HJM Derivation¶
The Hull-White model (1990) specifies the risk-neutral short rate dynamics:
where \(a > 0\) is the mean-reversion speed, \(\sigma > 0\) is the short-rate volatility, and \(\theta(t)\) is a time-dependent drift chosen to exactly fit the initial term structure. The model arises from the Heath-Jarrow-Morton framework (Chapter 19) by choosing a deterministic volatility function \(\sigma_f(t,T) = \sigma e^{-a(T-t)}\) for the instantaneous forward rate. The HJM drift condition then determines the risk-neutral drift endogenously: \(\theta(t) = \partial_t f(0,t) + af(0,t) + \frac{\sigma^2}{2a}(1-e^{-2at})\), where \(f(0,t)\) is the initial forward curve. Under HJM, the forward rate dynamics satisfy \(df(t,T) = \alpha(t,T)\,dt + \sigma e^{-a(T-t)}\,dW_t\) with drift \(\alpha(t,T) = \frac{\sigma^2}{a}e^{-a(T-t)}(1-e^{-a(T-t)})\), and the short rate is recovered as \(r_t = f(t,t)\). This is the extended Vasicek model: the constant \(\theta\) of the Vasicek model becomes \(\theta(t)\) to absorb the initial term structure, ensuring \(P^{\text{model}}(0,T) = P^{\text{market}}(0,T)\) for all \(T\)---a critical requirement for interest rate derivative pricing. The connection to Vasicek is precise: setting \(\theta(t) = a\theta_{\infty}\) recovers the constant-parameter Vasicek model \(dr_t = a(\theta_{\infty} - r_t)\,dt + \sigma\,dW_t\) with long-run mean \(\theta_{\infty}\), stationary distribution \(r_\infty \sim N(\theta_{\infty}, \sigma^2/(2a))\), and half-life \(t_{1/2} = \ln 2/a\). The affine structure means the model belongs to the class where bond prices take the exponential-affine form \(P(t,T) = e^{A(t,T) - B(t,T)r_t}\), with \(A\) and \(B\) satisfying Riccati-type ODEs \(B'(\tau) = 1 - aB(\tau)\) and \(A'(\tau) = -\theta(t)B(\tau) + \frac{1}{2}\sigma^2 B(\tau)^2\).
Short Rate Solution and Distribution¶
The Ornstein-Uhlenbeck SDE has the explicit solution:
derived by applying the integrating factor \(e^{at}\) and integrating. The short rate is Gaussian with conditional distribution \(r_t \mid r_s \sim N(\mu(s,t), \Sigma^2(s,t))\) where \(\mu(s,t) = r_s e^{-a(t-s)} + \int_s^t e^{-a(t-u)}\theta(u)\,du\) and \(\Sigma^2(s,t) = \frac{\sigma^2}{2a}(1-e^{-2a(t-s)})\), computed via the Ito isometry. The short rate decomposition \(r_t = \tilde{r}(t) + \psi(t)\) separates the stochastic zero-mean Gaussian process \(\psi(t)\) satisfying \(d\psi = -a\psi\,dt + \sigma\,dW_t\) with \(\psi(0)=0\) from the deterministic component \(\tilde{r}(t) = r_0 e^{-at} + \int_0^t e^{-a(t-u)}\theta(u)\,du\) absorbing \(\theta(t)\) and the initial curve. The characteristic function of \(\psi(t)\) is \(\phi_\psi(u,t) = \exp(iu\cdot 0 - \frac{1}{2}u^2 \sigma_r^2(t))\) where \(\sigma_r^2(t) = \frac{\sigma^2}{2a}(1-e^{-2at})\). By expressing \(\theta(t)\) in terms of the initial forward curve, the deterministic part satisfies \(\tilde{r}(t) = f(0,t) + \frac{\sigma^2}{2a^2}(1-e^{-at})^2\), enabling direct computation from market observables. The instantaneous forward rate under Hull-White satisfies \(f(t,T) = -\partial_T \ln P(t,T) = f(0,T) + B'(T-t)[r_t - f(0,t)] + \frac{\sigma^2}{2a^2}[(1-e^{-a(T-t)})^2 - (1-e^{-at})^2 e^{-2a(T-t)}/(e^{-2at})]\). Fitting \(\theta(t)\) to the yield curve is achieved analytically: \(\theta(t) = \partial_t f(0,t) + af(0,t) + \frac{\sigma^2}{2a}(1-e^{-2at})\), requiring only the initial forward curve and its derivative. A known limitation: the Gaussian distribution allows negative rates, which is acceptable for some currencies but problematic for others---the Black-Karasinski model \(d\ln r_t = [\theta(t) - a\ln r_t]\,dt + \sigma\,dW_t\) addresses this at the cost of losing analytical tractability.
Bond Pricing and the Affine Structure¶
The model's affine structure yields the exponential-affine bond price:
where \(B(t,T) = \frac{1-e^{-a(T-t)}}{a}\) and \(A(t,T)\) is determined by fitting to the initial curve: \(A(t,T) = \ln\frac{P(0,T)}{P(0,t)} - B(t,T)f(0,t) - \frac{\sigma^2}{4a}B(t,T)^2(1-e^{-2at})\). This can be derived either via the expectation method \(P(t,T) = \mathbb{E}^{\mathbb{Q}}[e^{-\int_t^T r_s\,ds} \mid r_t]\), exploiting the Gaussian distribution of \(\int_t^T r_s\,ds\) so that \(P(t,T) = \exp(-\mathbb{E}[\int_t^T r_s\,ds \mid r_t] + \frac{1}{2}\text{Var}[\int_t^T r_s\,ds \mid r_t])\) by the moment-generating function of a Gaussian, or via the PDE method with the bond pricing equation \(\partial_t P + (\theta - ar)\partial_r P + \frac{1}{2}\sigma^2\partial_{rr}P - rP = 0\) and the affine ansatz \(P = e^{A-Br}\) leading to the Riccati system. The ZCB dynamics are \(dP(t,T)/P(t,T) = r_t\,dt - \sigma B(t,T)\,dW_t\), showing that \(\sigma_P(t,T) = \sigma B(t,T)\) is the bond price volatility. Consistency with the initial yield curve is verified: \(P(0,T) = \exp(A(0,T) - B(0,T)r_0) = P^{\text{market}}(0,T)\) by construction. Yield curve dynamics under Hull-White preserve the parallel-shift-plus-decay structure: yields at all maturities are affine functions of \(r_t\), with loadings \(B(t,T)/(T-t)\) that decay exponentially, and \(y(t,T) = -A(t,T)/(T-t) + B(t,T)r_t/(T-t)\).
Named Functions and Measure Change¶
The named functions provide a compact notation for all Hull-White pricing formulas. The fundamental functions are: \(B(\tau) = (1-e^{-a\tau})/a\) (duration-like); \(\theta(t) = \partial_t f(0,t) + af(0,t) + \frac{\sigma^2}{2a}(1-e^{-2at})\) (drift); \(\tilde{r}(t) = f(0,t) + \frac{\sigma^2}{2a^2}(1-e^{-at})^2\) (deterministic mean); \(\sigma_r^2(t) = \frac{\sigma^2}{2a}(1-e^{-2at})\) (short rate variance); \(A(t,T) = \ln\frac{P(0,T)}{P(0,t)} - B(t,T)f(0,t) - \frac{\sigma^2}{4a}B(t,T)^2(1-e^{-2at})\) (bond price intercept). The characteristic function of the integrated short rate is \(\phi_{\int r}(u;t,T) = \exp\!\left(iu\,M(t,T) - \frac{1}{2}u^2 V(t,T)\right)\) where \(M(t,T) = \mathbb{E}^{\mathbb{Q}}[\int_t^T r_s\,ds \mid r_t]\) and \(V(t,T) = \text{Var}[\int_t^T r_s\,ds \mid r_t] = \frac{\sigma^2}{a^2}[\tau - 2B(\tau) + \frac{1}{2a}(1-e^{-2a\tau})]\), with \(\tau = T-t\). These functions satisfy interrelationships: \(B(\tau) \to \tau\) as \(a \to 0\), \(\sigma_r^2(t) \to \sigma^2 t\) as \(a \to 0\) (Brownian motion limit), and \(V(t,T) \to \frac{1}{3}\sigma^2\tau^3\) as \(a \to 0\) (Ho--Lee limit). The functions are derived via Riccati equations from the affine ansatz.
The \(T\)-forward measure \(\mathbb{Q}^T\), defined by the numeraire \(P(t,T)\) with Radon--Nikodym derivative \(dQ^T/dQ|_{\mathcal{F}_t} = P(t,T)/(B_t P(0,T))\), transforms the short rate drift: under \(\mathbb{Q}^T\), \(dr_t = [\theta(t) - ar_t - \sigma^2 B(t,T)]\,dt + \sigma\,dW_t^T\)---the drift is reduced by \(\sigma^2 B(t,T)\), reflecting the convexity adjustment. The Girsanov transformation gives \(dW_t^T = dW_t^{\mathbb{Q}} + \sigma B(t,T)\,dt\). Under the \(T\)-forward measure, the forward rate \(f(t,T)\) satisfies \(df(t,T) = \sigma e^{-a(T-t)}\,dW_t^T\) (driftless martingale), the bond price ratio \(P(t,S)/P(t,T)\) satisfies \(d(P(t,S)/P(t,T)) = \sigma[B(t,S) - B(t,T)] \cdot (P(t,S)/P(t,T))\,dW_t^T\) (also a martingale), and forward bond prices \(P(t,S)/P(t,T)\) are martingales under \(\mathbb{Q}^T\), simplifying the pricing of options on bonds.
Bond Options, Caps, and Swaptions¶
The zero-coupon bond option price has the Black-like formula:
where \(d_{\pm} = \frac{\ln(P(t,S)/(K\,P(t,T)))}{\sigma_P} \pm \frac{\sigma_P}{2}\) and \(\sigma_P = \frac{\sigma}{a}(1-e^{-a(S-T)})\sqrt{\frac{1-e^{-2a(T-t)}}{2a}}\) is the bond price volatility. The derivation proceeds under the \(T\)-forward measure where \(P(t,S)/P(t,T)\) is log-normal, reducing to the Black-Scholes structure. Jamshidian's trick decomposes a coupon bond option into a portfolio of zero-coupon bond options: since all bond prices \(P(T,T_i)\) are monotone decreasing functions of \(r_T\) in affine models, the coupon bond option exercises optimally at a unique threshold \(r^*\) solving \(\sum_i c_i P(T,T_i; r^*) = K\), and the payoff splits as \(\sum_i c_i(P(T,T_i) - K_i)^+\) where \(K_i = P_{r^*}(T,T_i) = \exp(A(T,T_i) - B(T,T_i)r^*)\). Each term is a zero-coupon bond option priced analytically.
Caplets are put options on zero-coupon bonds: Caplet(t; T_i-1, T_i) = (1+δ K_cap) Put(t; T_i-1, T_i, 1/(1+δ K_cap)) where δ = T_i - T_i-1, with the explicit Hull-White formula Caplet = (1+δ K) [KP(0,T_i-1)N(-d_-^cap) - P(0,T_i)N(-d_+^cap)]/(1+δ K) using d_±^cap = ln(P(0,T_i)/(P(0,T_i-1)·K))/σ_P ± σ_P/2 and K = 1/(1+δ K). Caps are portfolios of caplets: Cap = Σ_i=1ⁿ Caplet_i. Floorlets satisfy put-call parity for bond options. Swap pricing uses V_swap = N[P(t,T_0) - P(t,T_n) - KΣ_i=1ⁿ δ_i P(t,T_i)]. Swaptions are priced via Jamshidian's trick applied to the swap as a coupon bond: the payer swaption max(Σ c_i P(T_0,T_i) - 1, 0) with c_i = δ K for i < n and c_n = 1 + δ K decomposes into ZCB options, or alternatively by direct integration under the annuity measure, yielding a Black-like formula with the swap rate volatility σ_S = √Σ_i,j w_i w_j σ B(T_0,T_i)σ B(T_0,T_j)1-e^-2aT_0/2a where w_i = c_i P(0,T_i)/Σ_j c_j P(0,T_j). Bermudan swaptions require tree or Monte Carlo methods with backward induction and early-exercise comparison.¶
Calibration¶
Hull-White calibration proceeds in two stages: (1) the initial curve \(P(0,T)\) determines \(\theta(t)\) analytically via \(\theta(t) = \partial_t f(0,t) + af(0,t) + \frac{\sigma^2}{2a}(1-e^{-2at})\)---this is exact by construction; (2) the volatility parameters \((a, \sigma)\) are calibrated to cap volatilities or swaption volatilities by minimizing \(\sum_i w_i(\sigma_{\text{model}}^i - \sigma_{\text{market}}^i)^2\). Calibration to caps proceeds by matching model caplet volatilities \(\sigma_{\text{HW}}^{\text{cap}}(T_i) = \frac{\sigma}{a}(1-e^{-a\delta_i})\sqrt{\frac{1-e^{-2aT_{i-1}}}{2a}}\) to market Black volatilities at each maturity. Calibration to swaptions matches model swaption volatilities derived from Jamshidian's decomposition to market data across the expiry-tenor matrix. Joint calibration to both caps and swaptions is overconstrained for a two-parameter model, requiring trade-offs and the objective \(\min_{a,\sigma} \alpha\sum_i w_i^{\text{cap}}(e_i^{\text{cap}})^2 + (1-\alpha)\sum_j w_j^{\text{swap}}(e_j^{\text{swap}})^2\). The mean-reversion \(a\) primarily controls the decorrelation between short and long rates (and hence the swaption skew), while \(\sigma\) scales the overall volatility level. Parameter stability is important: small changes in \(a\) can significantly affect long-dated swaption prices while barely changing cap prices, creating instability in joint calibration. Regularization via \(\lambda\|(a,\sigma) - (a_{\text{prev}}, \sigma_{\text{prev}})\|^2\) and time-dependent extensions \(\sigma(t)\) piecewise constant improve stability and fit. Identifiability: \(a\) and \(\sigma\) are well-identified from at-the-money volatilities across multiple tenors, but the two-parameter model cannot fit the full volatility surface, motivating extensions.
Tree Implementation¶
The trinomial tree (Hull-White, 1994) discretizes the short rate process for pricing path-dependent and early-exercise derivatives. Construction: (1) build a tree for the zero-mean process \(x_t\) on a recombining grid with spacing \(\Delta x = \sigma\sqrt{3\Delta t}\); (2) choose branching probabilities to match the first two moments of \(x_t\), using normal, up, or down branching patterns depending on the node position relative to the center; (3) shift each time slice by \(\alpha(t_i)\) to fit the initial yield curve, where \(r_{ij} = \alpha_i + j\Delta x\). Arrow-Debreu prices \(Q_{ij}\) propagated through the tree enable efficient calibration: \(\alpha(t_i)\) is chosen so that \(\sum_j Q_{ij}e^{-(\alpha_i + j\Delta x)\Delta t} = P(0, t_{i+1})\), solved iteratively. Pricing derivatives uses backward induction: at each node, compute continuation value as the discounted probability-weighted average of successor node values, \(V_{ij} = e^{-r_{ij}\Delta t}\sum_k p_k V_{i+1,k}\). American and Bermudan swaptions are priced by backward induction with early exercise comparison \(V_{ij} = \max(V_{ij}^{\text{cont}}, V_{ij}^{\text{exercise}})\) at each exercise-eligible node. The tree naturally handles time-dependent \(\theta(t)\) and provides a lattice visualization of the yield curve dynamics.
Two-Factor Extension¶
The two-factor Hull-White model adds a second mean-reverting factor:
with \(d\langle W^{(1)}, W^{(2)}\rangle_t = \rho\,dt\). Equivalently, writing \(r(t) = x(t) + y(t) + \varphi(t)\) where \(dx = -ax\,dt + \sigma_1\,dW^{(1)}\), \(dy = -by\,dt + \sigma_2\,dW^{(2)}\), and \(\varphi(t)\) is deterministic, the short rate decomposes into two independent OU factors plus a drift. Bond prices remain exponential-affine: \(P(t,T) = \frac{P(0,T)}{P(0,t)}\exp\!\left(A^{(2)}(t,T) - B_a(\tau)x_t - B_b(\tau)y_t\right)\) where \(B_a(\tau) = (1-e^{-a\tau})/a\), \(B_b(\tau) = (1-e^{-b\tau})/b\), and \(A^{(2)}\) involves cross terms from the correlation. The additional factor \(u_t\) allows richer yield curve dynamics: twist and butterfly movements beyond the parallel shifts of the one-factor model. The correlation \(\rho\) between factors controls the shape of the yield curve volatility surface, with \(\rho < 0\) producing decorrelation between short and long ends. Calibration with four parameters \((a, b, \sigma_1, \sigma_2)\) plus \(\rho\) fits the swaption volatility matrix more accurately than the one-factor model, particularly for off-diagonal (non-co-terminal) swaptions.
Monte Carlo Simulation¶
Exact simulation exploits the Gaussian transition density: \(r_{t+\Delta t} = \mu(t, t+\Delta t) + \Sigma(t, t+\Delta t)\,Z\) with \(Z \sim N(0,1)\), requiring no discretization error since the OU transition is known in closed form. Bond prices along each path are computed via \(P(t_i, T) = \exp(A(t_i, T) - B(t_i, T)r_{t_i})\), also exact. Simulation of bond prices involves computing \(P(t_i, T_j)\) at each time step for all relevant maturities, enabling path-wise portfolio valuation. Path-dependent derivatives (e.g., callable bonds, range accruals, TARNs, auto-callable notes) are priced by simulating the short rate path and applying exercise/knock-out conditions at each monitoring date. Variance reduction techniques include: antithetic variates (pairing \(Z\) with \(-Z\) to halve variance for symmetric payoffs), control variates using the analytical bond price as a known-expectation control \(\hat{V} = \bar{V}_{\text{MC}} - \beta(\bar{C}_{\text{MC}} - C_{\text{exact}})\), stratified sampling on the initial short rate to reduce sampling variance, and importance sampling shifting the drift toward high-payoff regions.
Role in the Book
The Hull-White model is the workhorse short-rate model for interest rate derivatives, complementing the HJM and LIBOR market model frameworks (Chapter 19). Its affine structure connects to the general theory of Chapter 15, the calibration techniques parallel Chapter 17, and the tree and Monte Carlo implementations extend the numerical methods of Chapter 8 to the interest rate domain. The model's Gaussian limitation motivates the CIR-based and log-normal alternatives discussed in Chapter 18.